nonminimalMaps -- find the degree zero maps in the Schreyer resolution of an ideal

Usage:

(C, H) = nonminimalMaps I
Inputs:
- I, an ideal, in a polynomial ring $S$ over a base field or coefficient ring $A$. The lead terms of the generators of $I$ should be the initial ideal of $I$, and should be monic.
Outputs:
- C, a chain complex, A complex over a polynomial ring where any parameters in the base ring are set to have degree 0, and the variables of the ring of $I$ are set to have degree one.
- H, a hash table, Whose keys describe which submatrix in the resolution this is, and whose values are those submatrices (placed into the original coefficient ring $A$)

Description

The Schreyer resolution of $I$ (which is generally non-minimal) is computed. The nonminimal parts are the submatrices in this resolution which do not involve the variables in $S$. They are elements in the base ring $A$. For instance, H#(\ell, d) is the submatrix of the matrix from $C_{\ell+1} \to C_{\ell}$ sending degree $d$ to degree $d$.

The ranks of these matrices for a specific parameter value determine exactly the minimal Betti table for the ideal $I$, evaluated at that parameter point.

Now for our example.

i1 : kk = ZZ/101;

i2 : S = kk[a..d];

i3 : F = groebnerFamily ideal"a2,ab,ac,b2,bc2,c3"

             2                      2                      2               
o3 = ideal (a  + t b*c + t a*d + t c  + t b*d + t c*d + t d , a*b + t b*c +
                  1       3       2      4       5       6           7     
     ------------------------------------------------------------------------
                2                         2                              2  
     t a*d + t c  + t  b*d + t  c*d + t  d , a*c + t  b*c + t  a*d + t  c  +
      9       8      10       11       12           13       15       14    
     ------------------------------------------------------------------------
                           2   2                         2                  
     t  b*d + t  c*d + t  d , b  + t  b*c + t  a*d + t  c  + t  b*d + t  c*d
      16       17       18          19       21       20      22       23   
     ------------------------------------------------------------------------
           2     2                    2       2          2         2       3 
     + t  d , b*c  + t  b*c*d + t  a*d  + t  c d + t  b*d  + t  c*d  + t  d ,
        24            25         27        26       28        29        30   
     ------------------------------------------------------------------------
      3                    2       2          2         2       3
     c  + t  b*c*d + t  a*d  + t  c d + t  b*d  + t  c*d  + t  d )
           31         33        32       34        35        36

o3 : Ideal of kk[t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  , t  , t , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  ][a..d]
                  6   12   5   30   18   4   24   36   11   2   29   3   10   17   1   23   28   35   8   16   9   22   26   34   7   14   27   15   20   25   32   13   21   33   19   31

i4 : (C, H) = nonminimalMaps F;

i5 : betti(C, Weights => {1,1,1,1})

            0 1  2 3 4
o5 = total: 1 6 10 6 1
         0: 1 .  . . .
         1: . 4  4 2 .
         2: . 2  5 3 1
         3: . .  1 1 .

o5 : BettiTally

We see that there are 4 maps that are nonminimal (of sizes $2 \times 4$, $5 \times 2$, $1 \times 3$, and $1 \times 1$).

i6 : keys H

o6 = {(3, 4), (3, 5), (4, 6), (2, 3)}

o6 : List

i7 : H#(2,3)

o7 = {3} | -t_8-t_20t_13      t_7t_20-t_14t_20+t_20t_13t_19            
     {3} | -t_7+t_14-t_13t_19 -t_8-t_20t_13+t_7t_19-t_14t_19+t_13t_19^2
     ------------------------------------------------------------------------
     -t_2-t_14^2+t_20t_13^2    -t_8t_14+t_1t_20+t_7t_20t_13             |
     -t_1-2t_14t_13+t_13^2t_19 -t_2-t_7t_14-t_8t_13+t_1t_19+t_7t_13t_19 |

                                                                                                                                                                                           2                                                                                                                                                                                    4
o7 : Matrix (kk[t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  , t  , t , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  ])  <-- (kk[t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  , t  , t , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  ])
                 6   12   5   30   18   4   24   36   11   2   29   3   10   17   1   23   28   35   8   16   9   22   26   34   7   14   27   15   20   25   32   13   21   33   19   31             6   12   5   30   18   4   24   36   11   2   29   3   10   17   1   23   28   35   8   16   9   22   26   34   7   14   27   15   20   25   32   13   21   33   19   31

i8 : H#(3,4)

o8 = {4} | -t_20                                   
     {4} | -1                                      
     {4} | t_8+t_20t_13-t_7t_19+t_14t_19-t_13t_19^2
     {4} | -t_7+t_14-t_13t_19                      
     {4} | 0                                       
     ------------------------------------------------------------------------
     -t_8                                    |
     t_13                                    |
     t_2+t_7t_14+t_8t_13-t_1t_19-t_7t_13t_19 |
     -t_1-2t_14t_13+t_13^2t_19               |
     t_7-t_14+t_13t_19                       |

                                                                                                                                                                                           5                                                                                                                                                                                    2
o8 : Matrix (kk[t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  , t  , t , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  ])  <-- (kk[t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  , t  , t , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  ])
                 6   12   5   30   18   4   24   36   11   2   29   3   10   17   1   23   28   35   8   16   9   22   26   34   7   14   27   15   20   25   32   13   21   33   19   31             6   12   5   30   18   4   24   36   11   2   29   3   10   17   1   23   28   35   8   16   9   22   26   34   7   14   27   15   20   25   32   13   21   33   19   31

i9 : H#(3,5)

o9 = {5} | -1 t_13 -t_14 |

                                                                                                                                                                                           1                                                                                                                                                                                    3
o9 : Matrix (kk[t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  , t  , t , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  ])  <-- (kk[t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  , t  , t , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  ])
                 6   12   5   30   18   4   24   36   11   2   29   3   10   17   1   23   28   35   8   16   9   22   26   34   7   14   27   15   20   25   32   13   21   33   19   31             6   12   5   30   18   4   24   36   11   2   29   3   10   17   1   23   28   35   8   16   9   22   26   34   7   14   27   15   20   25   32   13   21   33   19   31

i10 : H#(4,6)

o10 = {6} | -1 |

                                                                                                                                                                                            1                                                                                                                                                                                    1
o10 : Matrix (kk[t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  , t  , t , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  ])  <-- (kk[t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  , t  , t , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  ])
                  6   12   5   30   18   4   24   36   11   2   29   3   10   17   1   23   28   35   8   16   9   22   26   34   7   14   27   15   20   25   32   13   21   33   19   31             6   12   5   30   18   4   24   36   11   2   29   3   10   17   1   23   28   35   8   16   9   22   26   34   7   14   27   15   20   25   32   13   21   33   19   31

Let's impose the condition that the map H#(2,3) vanishes (so has rank 0). The Betti diagram of such ideals is not the one for a set of 6 generic points in $\PP^3$.

i11 : J = trim(minors(1, H#(2,3)) + groebnerStratum F);

o11 : Ideal of kk[t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  , t , t  , t  , t  , t , t  , t , t  , t  , t  , t , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  , t  ]
                   6   12   5   30   18   4   24   36   11   2   29   3   10   17   1   23   28   35   8   16   9   22   26   34   7   14   27   15   20   25   32   13   21   33   19   31

i12 : compsJ = decompose J;

i13 : #compsJ

o13 = 2

i14 : pt1 = randomPointOnRationalVariety compsJ_0

o14 = | -6 48 44 -23 -2 -11 -35 -26 27 -43 48 27 15 -22 25 -16 34 -29 46 -20
      -----------------------------------------------------------------------
      40 21 -30 -38 -19 -8 -36 39 19 -29 -16 -29 -10 19 24 -24 |

               1       36
o14 : Matrix kk  <-- kk

i15 : pt2 = randomPointOnRationalVariety compsJ_1

o15 = | -48 -46 16 17 -1 -43 15 -1 12 -18 -6 -28 14 -28 -9 32 -22 -39 6 -47
      -----------------------------------------------------------------------
      28 -37 -47 38 -16 -15 34 27 -13 -43 22 16 0 -18 19 2 |

               1       36
o15 : Matrix kk  <-- kk

i16 : F1 = sub(F, (vars S)|pt1)

              2              2                             2               
o16 = ideal (a  + 25b*c - 43c  + 27a*d - 11b*d + 44c*d - 6d , a*b - 19b*c +
      -----------------------------------------------------------------------
         2                              2                  2                
      46c  + 40a*d + 15b*d + 27c*d + 48d , a*c - 29b*c - 8c  + 39a*d - 20b*d
      -----------------------------------------------------------------------
                  2   2              2                              2     2  
      - 22c*d - 2d , b  + 24b*c + 19c  - 10a*d + 21b*d - 16c*d - 35d , b*c  -
      -----------------------------------------------------------------------
                   2         2        2        2      3   3                2 
      29b*c*d - 30c d - 36a*d  + 34b*d  + 48c*d  - 23d , c  - 24b*c*d - 16c d
      -----------------------------------------------------------------------
             2        2        2      3
      + 19a*d  - 38b*d  - 29c*d  - 26d )

o16 : Ideal of S

i17 : betti res F1

             0 1 2 3
o17 = total: 1 6 8 3
          0: 1 . . .
          1: . 4 4 1
          2: . 2 4 2

o17 : BettiTally

i18 : F2 = sub(F, (vars S)|pt2)

              2             2                              2               
o18 = ideal (a  - 9b*c - 18c  - 28a*d - 43b*d + 16c*d - 48d , a*b - 16b*c +
      -----------------------------------------------------------------------
        2                              2                   2                
      6c  + 28a*d + 14b*d + 12c*d - 46d , a*c + 16b*c - 15c  + 27a*d - 47b*d
      -----------------------------------------------------------------------
                 2   2              2                      2     2          
      - 28c*d - d , b  + 19b*c - 13c  - 37b*d + 32c*d + 15d , b*c  - 43b*c*d
      -----------------------------------------------------------------------
           2         2        2       2      3   3               2         2
      - 47c d + 34a*d  - 22b*d  - 6c*d  + 17d , c  + 2b*c*d + 22c d - 18a*d 
      -----------------------------------------------------------------------
             2        2    3
      + 38b*d  - 39c*d  - d )

o18 : Ideal of S

i19 : betti res F2

             0 1 2 3
o19 = total: 1 6 8 3
          0: 1 . . .
          1: . 4 4 1
          2: . 2 4 2

o19 : BettiTally

What are the ideals F1 and F2?

i20 : netList decompose F1

      +---------------------------------------------------------------------------------------------------------------------------------------------------------+
o20 = |ideal (c + 39d, b + 27d, a - 18d)                                                                                                                        |
      +---------------------------------------------------------------------------------------------------------------------------------------------------------+
      |                            2              2                     2   3                2        2        2      3     2                2         2      3 |
      |ideal (a - 29b - 8c - 13d, b  + 24b*c + 19c  + 34b*d + 5c*d + 37d , c  - 24b*c*d - 16c d + 8b*d  + 22c*d  + 19d , b*c  - 29b*c*d - 30c d - 38c*d  + 14d )|
      +---------------------------------------------------------------------------------------------------------------------------------------------------------+

i21 : netList decompose F2

      +-------------------------------------------------------+
o21 = |ideal (c - 32d, b - 5d, a - 29d)                       |
      +-------------------------------------------------------+
      |ideal (c + 43d, b - 47d, a - 27d)                      |
      +-------------------------------------------------------+
      |ideal (c + 24d, b - 49d, a)                            |
      +-------------------------------------------------------+
      |ideal (c + 14d, b + 31d, a - 16d)                      |
      +-------------------------------------------------------+
      |                                      2              2 |
      |ideal (b + 11c + 22d, a + 11c + 42d, c  - 43c*d + 31d )|
      +-------------------------------------------------------+

We can determine what these represent. One should be a set of 6 points, where 5 lie on a plane. The other should be 6 points with 3 points on one line, and the other 3 points on a skew line.

Ways to use nonminimalMaps:

nonminimalMaps(Ideal)

For the programmer

The object nonminimalMaps is a method function.

The source of this document is in GroebnerStrata.m2:1012:0.

nonminimalMaps -- find the degree zero maps in the Schreyer resolution of an ideal

Description

See also

Ways to use nonminimalMaps:

For the programmer